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Chapter 5: Problem 45

Verify the identity. $$\sin t \csc \left(\frac{\pi}{2}-t\right)=\tan t$$

Short Answer

Expert verified
Yes, the given trigonometric identity \(\sin t \csc \left(\frac{\pi}{2}-t\right)=\tan t\) holds true.

Step by step solution

01

Write in terms of basic trigonometric functions

Recall that \(\csc x = \frac{1}{\sin x}\). Using this, the left side becomes: \(\sin t \cdot \frac{1}{\sin \left(\frac{\pi}{2}-t\right)}\).
02

Apply sine transformation

Next, since \(\sin(\frac{\pi}{2}-t) = cos(t)\), the left side expression can be rewritten as: \(\sin t \cdot \frac{1}{\cos t}\).
03

Simplify the expression

The expression \(\sin t \cdot \frac{1}{\cos t}\) simplifies to \(\tan t\), which is the right side of the given equation. Therefore, the given identity is verified.

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